The Mirror reports that Astori was on the shopping list before Dejan Lovren joined the Saints in January.
The 26-year-old has also been strongly linked with the likes of AC Milan and Everton.
The £10million-rated defender played for Italy in the Confederations Cup in Brazil last summer.
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Chelsea legend Didier Drogba has revealed his top 11 Chelsea career moments during his recent interview on Chelsea TV.
The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.